Rational ArithmeticMathematicaFunctions to Evaluate the One-sided One-sample K-S Cumulative Sampling Distribution

Abstract

One of the most widely used goodness-of-fit tests is the Kolmogorov-Smirnov (K-S) family of tests which have been implemented by many computer statistical software packages. To calculate a p value (evaluate the cumulative sampling distribution), these packages use various methods including recursion formulae, limiting distributions, and approximations of unknown accuracy developed over thirty years ago. Based on an extensive literature search for the one-sided one-sample K-S test, this paper identifies two direct formulae and five recursion formulae that can be used to calculate a p value and then develops two additional direct formulae and four iterative versions of the direct formulae for a total of thirteen formulae. To ensure accurate calculation by avoiding catastrophic cancelation and eliminating rounding error, each formula is implemented in rational arithmetic. Linear search is used to calculate the inverse of the cumulative sampling distribution (find the confidence interval bandwidth). Extensive tables of bandwidths are presented for sample sizes up to 2, 000. The results confirm the hypothesis that as the number of digits in the numerator and denominator integers of the rational number test statistic increases, the computation time also increases. In comparing the computational times of the thirteen formulae, the direct formulae are slightly faster than their iterative versions and much faster than all the recursion formulae. Computational times for the fastest formula are given for sample sizes up to fifty thousand.

Arbitrary PrecisionMathematicaFunctions to Evaluate the One-Sided One Sample K-S Cumulative Sampling Distribution

“…Arbitrary precision methods are used to calculate one-sided one sample K-S p values where the accuracy of the resultant p value is specified by the user. Comparative analysis of the rational arithmetic implementations in Brown and Harvey (2007) with the arbitrary precision implementations in this paper show that the arbitrary precision methods are over ten times faster than the rational arithmetic methods. Although arbitrary precision methods are faster, a major difficulty is determining the accuracy of any p value produced by arbitrary precision methods.…”

“…In a recent paper, Brown and Harvey (2007) evaluated 13 formulae for calculating the exact p values of the one-sided one sample Kolmogorov-Smirnov (K-S) test. They used rational arithmetic so that the p values were calculated exactly and the implementation of all 13 formulae yielded the same p values.…”

“…Although arbitrary precision methods are faster, a major difficulty is determining the accuracy of any p value produced by arbitrary precision methods. The p values calculated by rational arithmetic in Brown and Harvey (2007) will be used to check the accuracy of this paper's arbitrary precision calculations.…”

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Efficient rational arithmetic methods that can exactly evaluate the cumulative sampling distribution of the one-sided one sample Kolmogorov-Smirnov (K-S) test have been developed by Brown and Harvey (2007) for sample sizes n up to fifty thousand. This paper implements in arbitrary precision the same 13 formulae to evaluate the one-sided one sample K-S cumulative sampling distribution.

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Rational ArithmeticMathematicaFunctions to Evaluate the Two-Sided One Sample K-S Cumulative Sampling Distribution

“…Indeed, many computer statistical software packages that implement both the two-sided and one-sided one sample K-S test use different methods to calculate the p values. Brown and Harvey (2007) have summarized how various computer statistical software packages calculate the p values for the one sample K-S tests. Since they have investigated the one-sided one sample case, this paper examines the two-sided one sample case.…”